{"paper":{"title":"The largest $(k, \\ell)$-sum-free sets in compact abelian groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.NT"],"primary_cat":"math.CO","authors_text":"Noah Kravitz","submitted_at":"2019-01-10T15:52:01Z","abstract_excerpt":"A subset $A$ of a finite abelian group is called $(k,\\ell)$-sum-free if $kA \\cap \\ell A=\\emptyset.$ In this paper, we extend this concept to compact abelian groups and study the question of how large a measurable $(k,\\ell)$-sum-free set can be. For integers $1 \\leq k <\\ell$ and a compact abelian group $G$, let $$\\lambda_{k,\\ell}(G)=\\sup\\{ \\mu(A): kA \\cap \\ell A =\\emptyset \\}$$ be the maximum possible size of a $(k,\\ell)$-sum-free subset of $G$. We prove that if $G=\\mathbb{I} \\times M$, where $\\mathbb{I}$ is the identity component of $G$, then $$\\lambda_{k, \\ell}(G)=\\max \\left\\{ \\lambda_{k, \\el"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.03233","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}